Viviani, Vincenzo, De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

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            <s xml:id="echoid-s9376" xml:space="preserve">Sumpto enim in data recta A B quocunque alio puncto H, vel in ipſius
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            parte producta vltra B, vt in prima figura, vel in ipſa A B, vt in ſecunda,
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            & </s>
            <s xml:id="echoid-s9377" xml:space="preserve">ex H ducta H I perpendiculari ad A B, ſecante diagonalem D B in I,
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            ductaque A I ſecante circuli peripheriam in L, iunctiſque G L, G I: </s>
            <s xml:id="echoid-s9378" xml:space="preserve">erit
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            angulus A L G rectus, atque externus trianguli L I G; </s>
            <s xml:id="echoid-s9379" xml:space="preserve">quare internus L I
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            G acutus erit, ac ideo recta I M, quæ ex I erigitur perpendicularis ad I A,
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            hoc eſt, quæ ipſi L G æquidiſtat, ſecabit A B vltra punctum G, vt in M, ac
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            ideo erit A G minor A M. </s>
            <s xml:id="echoid-s9380" xml:space="preserve">Et cum in triangulo rectangulo A I M, ſit vt A
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            H ad H I, ita H I ad H M, ſitque H I æqualis H B, erit A H ad H B, vt
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            H B ad H M, ergo A M eſt aggregatum extremarum proportionalium poſt
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            partes A H, H B, ſed eſt A G minor A M, vt modò oſtendimus: </s>
            <s xml:id="echoid-s9381" xml:space="preserve">ergo ag-
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            gregatum A G minus eſt aggregato A M: </s>
            <s xml:id="echoid-s9382" xml:space="preserve">& </s>
            <s xml:id="echoid-s9383" xml:space="preserve">hoc ſemper vbicunque aſſum-
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            ptum fuerit punctum H extra C: </s>
            <s xml:id="echoid-s9384" xml:space="preserve">ergo aggregatum A G minus eſt aggrega-
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            to A M: </s>
            <s xml:id="echoid-s9385" xml:space="preserve">& </s>
            <s xml:id="echoid-s9386" xml:space="preserve">hoc ſemper vbicunque aſſumptum fuerit punctum H extra C:
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            </s>
            <s xml:id="echoid-s9387" xml:space="preserve">quare A G eſt _MINIMVM_ aggregatum quæſitum; </s>
            <s xml:id="echoid-s9388" xml:space="preserve">& </s>
            <s xml:id="echoid-s9389" xml:space="preserve">recta A B ſecta eſt in
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            C, vt imperatum fuit. </s>
            <s xml:id="echoid-s9390" xml:space="preserve">Quod faciendum erat.</s>
            <s xml:id="echoid-s9391" xml:space="preserve"/>
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          <figure number="268">
            <image file="0337-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0337-01"/>
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        <div xml:id="echoid-div972" type="section" level="1" n="393">
          <head xml:id="echoid-head404" xml:space="preserve">SCHOLIVM.</head>
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            <s xml:id="echoid-s9392" xml:space="preserve">SI quæratur iuxta quam rationem repertum punctum C diuidat datam A
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            B; </s>
            <s xml:id="echoid-s9393" xml:space="preserve">id ex ipſa Theorematis conſtructione elicietur. </s>
            <s xml:id="echoid-s9394" xml:space="preserve">Nam cum triangu-
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            la D A B, B E F ſint ſimilia inter ſe, erit B D ad D A, ſiue diameter qua-
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            drati ad latus, vt B F ad F E, vel ad F A, & </s>
            <s xml:id="echoid-s9395" xml:space="preserve">cum ſit B C ad C E, vt C E
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            ad C F, ſitque B C æqualis C E (cum & </s>
            <s xml:id="echoid-s9396" xml:space="preserve">B A æqualis ſit A D) erit etiam
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            C E ſiue C B æqualis C F. </s>
            <s xml:id="echoid-s9397" xml:space="preserve">Quare ſi data recta B A diuidatur, ita vt pars
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            B F ad reliquam partem F A, ſit vt diameter cuiuſdam quadrati ad eius la-
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            tus, & </s>
            <s xml:id="echoid-s9398" xml:space="preserve">maior pars B F ſecetur bifariam in C, hoc ipſum punctum erit quæ-
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            ſitum.</s>
            <s xml:id="echoid-s9399" xml:space="preserve"/>
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            <s xml:id="echoid-s9400" xml:space="preserve">Vel. </s>
            <s xml:id="echoid-s9401" xml:space="preserve">Cum rectæ A B, A D ſint æquales, & </s>
            <s xml:id="echoid-s9402" xml:space="preserve">perpendiculariter conſtitu-
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            tæ, erit A D, ſiue D E latus quadrati, & </s>
            <s xml:id="echoid-s9403" xml:space="preserve">D B diameter, & </s>
            <s xml:id="echoid-s9404" xml:space="preserve">E B exceſſus
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            diametri ſuper latus, ſed eſt A C ad C B, vt D E ad E B: </s>
            <s xml:id="echoid-s9405" xml:space="preserve">ergo quæſitum
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            punctum C ſecat datam rectam A B, ita vt maior pars A C ad minorem C
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            B, ſit vt latus cuiuſdam quadrati ad exceſſum diametri ſuper latus, quæ ra-
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            tio, vt iam conſtat, cadit inter terminos incommenſurabiles.</s>
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